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# Parametric Equations: Cycloids

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The path of a general cycloid is parameterized by $$x=r(t-\sin{t})$$ and $$y=r(1-\cos{t})$$

(a) Find the length of one arch of the cycloid.

(b) Let $$L$$ be the length you found in part (a). One end $$A$$ of a string of length $$\dfrac{L}{2}$$ is tied at $$(0, 0)$$, and the other end $$B$$ is at $$(\pi r, 2r)$$, wrapping around one-half of one arch of the cycloid.

Prove that as the string is unwrapped, the point $$B$$ traces a path that is parameterized by $$x=r(t+\sin{t})$$ and $$y=r(3+\cos{t})$$. (This path is in fact congruent to an arch of the original cycloid.)

Hi, it would really help if anyone could help with part (b) as I have no idea where to start. For part (a) I got 8r for length L.

Jul 21, 2022